6 098 Digital and Computational Photography 6 882 Advanced Computational Photography Panoramas Bill Freeman Fr do Durand MIT EECS Lots of slides stolen from Alyosha Efros who stole them from Steve Seitz and Rick Szeliski Olivier Gondry Director of music video and commercial Special effect specialist Morphing rotoscoping Today at 5 40pm in 32 141 Why Mosaic Are you getting the whole picture Compact Camera FOV 50 x 35 Slide from Brown Lowe Why Mosaic Are you getting the whole picture Compact Camera FOV 50 x 35 Human FOV 200 x 135 Slide from Brown Lowe Why Mosaic Are you getting the whole picture Compact Camera FOV 50 x 35 Human FOV 200 x 135 Panoramic Mosaic 360 x 180 Slide from Brown Lowe Mosaics stitching images together virtual wide angle camera How to do it Basic Procedure Take a sequence of images from the same position Rotate the camera about its optical center Compute transformation between second image and first Transform the second image to overlap with the first Blend the two together to create a mosaic If there are more images repeat but wait why should this work at all What about the 3D geometry of the scene Why aren t we using it A pencil of rays contains all views real camera synthetic camera Can generate any synthetic camera view as long as it has the same center of projection Aligning images translation left on top right on top Translations are not enough to align the images Image reprojection mosaic PP The mosaic has a natural interpretation in 3D The images are reprojected onto a common plane The mosaic is formed on this plane Mosaic is a synthetic wide angle camera Image reprojection Basic question How to relate 2 images from same camera center how to map a pixel from PP1 to PP2 PP2 Answer Cast a ray through each pixel in PP1 Draw the pixel where that ray intersects PP2 PP1 But don t we need to know the geometry of the two planes in respect to the eye Observation Rather than thinking of this as a 3D reprojection think of it as a 2D image warp from one image to another Back to Image Warping Which t form is the right one for warping PP1 into PP2 e g translation Euclidean affine projective Translation Affine Perspective 2 unknowns 6 unknowns 8 unknowns Homography Projective mapping between any two PPs with the same center of projection rectangle should map to arbitrary quadrilateral parallel lines aren t but must preserve straight lines same as project rotate reproject called Homography wx x wy y w 1 To apply a homography H p H p Compute p Hp regular matrix multiply Convert p from homogeneous to image coordinates PP2 PP1 1D homogeneous coordinates Add one dimension to make life simpler x w represent point x w w w 1 x 1D homography Reproject to different line w w 1 x 1D homography Reproject to different line w w 1 x 1D homography Reproject to different line Equivalent to rotating 2D points reprojection is linear in homogeneous coordinates w w 1 x Same in 2D Reprojection homography 3x3 matrix wx x wy y w 1 H p p PP2 PP1 Image warping with homographies image plane in front image plane below black area where no pixel maps to Digression perspective correction From Photography London et al From Photography London et al From Photography London et al From Photography London et al Tilt shift lens 35mm SLR version Photoshop version perspective crop you control reflection and perspective independently Back to Image rectification p p To unwarp rectify an image Find the homography H given a set of p and p pairs How many correspondences are needed Tricky to write H analytically but we can solve for it Find such H that best transforms points p into p Use least squares Least Squares Example Say we have a set of data points X1 X1 X2 X2 X3 X3 etc e g person s height vs weight We want a nice compact formula line to predict X s from Xs Xa b X We want to find a and b How many X X pairs do we need 1 X a X a b X X 1 1 1 1 Ax B X 2a b X 2 X2 1 b What if the data is noisy X1 X 2 X3 1 1 a 1 b X 1 X2 X 3 overconstrained min Ax B 2 X2 Solving for homographies p Hp wx a wy d w g b e h c x f y i 1 Can set scale factor i 1 So there are 8 unkowns Set up a system of linear equations Ah b where vector of unknowns h a b c d e f g h T Note we do not know w but we can compute it from x y w gx hy 1 The equations are linear in the unknown Solving for homographies p Hp wx a wy d w g b e h c x f y i 1 Can set scale factor i 1 So there are 8 unkowns Set up a system of linear equations Ah b where vector of unknowns h a b c d e f g h T Need at least 8 eqs but the more the better Solve for h If overconstrained solve using least squares 2 min Ah b Can be done in Matlab using command see help lmdivide Panoramas 1 Pick one image red 2 Warp the other images towards it usually one by one 3 blend Recap Panorama reprojection 3D rotation homography Homogeneous coordinates are kewl Use feature correspondence Solve least square problem Se of linear equations Warp all images to a reference one Use your favorite blending changing camera center Does it still work synthetic PP PP1 PP2 Nodal point http www reallyrightstuff com pano index html Planar mosaic Cool applications of homographies Oh Durand Dorsey Limitations of 2D Clone Brushing Distortions due to foreshortening and surface orientation Clone brush Photoshop Click on a reference pixel blue Then start painting somewhere else Copy pixel color with a translation Perspective clone brush Oh Durand Dorsey unpublished Correct for perspective And other tricks Rotational Mosaics Can we say something more about rotational mosaics i e can we further constrain our H 3D 2D Perspective Projection Xc Yc Zc f uc u K 3D Rotation Model Projection equations 1 Project from image to 3D ray x0 y0 z0 u0 uc v0 vc f 2 Rotate the ray by camera motion x1 y1 z1 R01 x0 y0 z0 3 Project back into new source image u1 v1 fx1 z1 uc fy1 z1 vc Therefore x y z R x y z f u v f u v f H K0R01K1 1 Our homography has only 3 4 or 5 DOF depending if focal length is known same or different This makes image registration much better behaved Pairwise alignment Procrustes Algorithm Golub VanLoan Given two sets of matching points compute R pi R pi with 3D rays pi N xi yi zi N ui uc vi vc f A i pi pi …